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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.nc_beta_dist"></a><a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral
        Beta Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">non_central_beta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">non_central_beta_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">non_central_beta_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">non_central_beta</span><span class="special">;</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">non_central_beta_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
   <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>

   <span class="comment">// Constructor:</span>
   <span class="identifier">non_central_beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>

   <span class="comment">// Accessor to shape parameters:</span>
   <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
   <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>

   <span class="comment">// Accessor to non-centrality parameter lambda:</span>
   <span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
          The noncentral beta distribution is a generalization of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta
          Distribution</a>.
        </p>
<p>
          It is defined as the ratio
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">X = χ<sub>m</sub><sup>2</sup>(λ) / (χ<sub>m</sub><sup>2</sup>(λ) + χ<sub>n</sub><sup>2</sup>)</span>
          </p></blockquote></div>
<p>
          where <span class="serif_italic">χ<sub>m</sub><sup>2</sup>(λ)</span> is a noncentral <span class="serif_italic">χ<sup>2</sup></span> random variable with <span class="emphasis"><em>m</em></span>
          degrees of freedom, and χ<sub>n</sub><sup>2</sup>
is a central <span class="serif_italic">χ<sup>2</sup> </span>
          random variable with <span class="emphasis"><em>n</em></span> degrees of freedom.
        </p>
<p>
          This gives a PDF that can be expressed as a Poisson mixture of beta distribution
          PDFs:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span>

          </p></blockquote></div>
<p>
          where P(i;λ/2) is the discrete Poisson probability at <span class="emphasis"><em>i</em></span>,
          with mean λ/2, and I<sub>x</sub><sup>'</sup>(α, β) is the derivative of the incomplete beta function.
          This leads to the usual form of the CDF as:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref2.svg"></span>

          </p></blockquote></div>
<p>
          The following graph illustrates how the distribution changes for different
          values of λ:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/nc_beta_pdf.svg" align="middle"></span>

          </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.member_functions"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">non_central_beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
</pre>
<p>
          Constructs a noncentral beta distribution with shape parameters <span class="emphasis"><em>a</em></span>
          and <span class="emphasis"><em>b</em></span> and non-centrality parameter <span class="emphasis"><em>lambda</em></span>.
        </p>
<p>
          Requires a &gt; 0, b &gt; 0 and lambda &gt;= 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the parameter <span class="emphasis"><em>a</em></span> from which this object was
          constructed.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the parameter <span class="emphasis"><em>b</em></span> from which this object was
          constructed.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the parameter <span class="emphasis"><em>lambda</em></span> from which this object
          was constructed.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.non_member_accessors"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.non_member_accessors">Non-member
          Accessors</a>
        </h5>
<p>
          Most of the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member
          accessor functions</a> are supported: <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
          Distribution Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
          Density Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
        </p>
<p>
          Mean and variance are implemented using hypergeometric pfq functions and
          relations given in <a href="http://reference.wolfram.com/mathematica/ref/NoncentralBetaDistribution.html" target="_top">Wolfram
          Noncentral Beta Distribution</a>.
        </p>
<p>
          However, the following are not currently implemented: <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a> and
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>.
        </p>
<p>
          The domain of the random variable is [0, 1].
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.accuracy"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>)
          found on various platforms with various floating point types. The failures
          in the comparison to the <a href="http://www.r-project.org/" target="_top">R Math
          library</a>, seem to be mostly in the corner cases when the probability
          would be very small. Unless otherwise specified any floating-point type
          that is narrower than the one shown will have <a class="link" href="../../relative_error.html#math_toolkit.relative_error.zero_error">effectively
          zero error</a>.
        </p>
<div class="table">
<a name="math_toolkit.dist_ref.dists.nc_beta_dist.table_non_central_beta_CDF"></a><p class="title"><b>Table 5.4. Error rates for non central beta CDF</b></p>
<div class="table-contents"><table class="table" summary="Error rates for non central beta CDF">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                </th>
<th>
                  <p>
                    GNU C++ version 7.1.0<br> linux<br> double
                  </p>
                </th>
<th>
                  <p>
                    GNU C++ version 7.1.0<br> linux<br> long double
                  </p>
                </th>
<th>
                  <p>
                    Sun compiler version 0x5150<br> Sun Solaris<br> long double
                  </p>
                </th>
<th>
                  <p>
                    Microsoft Visual C++ version 14.1<br> Win32<br> double
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    Non Central Beta, medium parameters
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 0.998ε (Mean = 0.0649ε)</span><br>
                    <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
                    = 1.46e+26ε (Mean = 3.5e+24ε) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_Rmath_3_2_3_Non_Central_Beta_medium_parameters">And
                    other failures.</a>)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 824ε (Mean = 27.4ε)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 832ε (Mean = 38.1ε)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 242ε (Mean = 31ε)</span>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    Non Central Beta, large parameters
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 1.18ε (Mean = 0.175ε)</span><br>
                    <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
                    = 1.01e+36ε (Mean = 1.19e+35ε) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_Rmath_3_2_3_Non_Central_Beta_large_parameters">And
                    other failures.</a>)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 2.5e+04ε (Mean = 3.78e+03ε)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 2.57e+04ε (Mean = 4.45e+03ε)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 3.66e+03ε (Mean = 500ε)</span>
                  </p>
                </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.dist_ref.dists.nc_beta_dist.table_non_central_beta_CDF_complement"></a><p class="title"><b>Table 5.5. Error rates for non central beta CDF complement</b></p>
<div class="table-contents"><table class="table" summary="Error rates for non central beta CDF complement">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                </th>
<th>
                  <p>
                    GNU C++ version 7.1.0<br> linux<br> double
                  </p>
                </th>
<th>
                  <p>
                    GNU C++ version 7.1.0<br> linux<br> long double
                  </p>
                </th>
<th>
                  <p>
                    Sun compiler version 0x5150<br> Sun Solaris<br> long double
                  </p>
                </th>
<th>
                  <p>
                    Microsoft Visual C++ version 14.1<br> Win32<br> double
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    Non Central Beta, medium parameters
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 0.998ε (Mean = 0.0936ε)</span><br>
                    <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
                    = 7.5e+97ε (Mean = 1.37e+96ε) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_complement_Rmath_3_2_3_Non_Central_Beta_medium_parameters">And
                    other failures.</a>)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 396ε (Mean = 50.7ε)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 554ε (Mean = 57.2ε)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 624ε (Mean = 62.7ε)</span>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    Non Central Beta, large parameters
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 0.986ε (Mean = 0.188ε)</span><br>
                    <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
                    = +INFε (Mean = +INFε) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_complement_Rmath_3_2_3_Non_Central_Beta_large_parameters">And
                    other failures.</a>)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 6.83e+03ε (Mean = 993ε)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 3.56e+03ε (Mean = 707ε)</span>
                  </p>
                </td>
<td>
                  <p>
                    <span class="blue">Max = 1.25e+04ε (Mean = 1.49e+03ε)</span>
                  </p>
                </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
          Error rates for the PDF, the complement of the CDF and for the quantile
          functions are broadly similar.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.tests"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.tests">Tests</a>
        </h5>
<p>
          There are two sets of test data used to verify this implementation: firstly
          we can compare with a few sample values generated by the <a href="http://www.r-project.org/" target="_top">R
          library</a>. Secondly, we have tables of test data, computed with this
          implementation and using interval arithmetic - this data should be accurate
          to at least 50 decimal digits - and is the used for our accuracy tests.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h4"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.implementation"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.implementation">Implementation</a>
        </h5>
<p>
          The CDF and its complement are evaluated as follows:
        </p>
<p>
          First we determine which of the two values (the CDF or its complement)
          is likely to be the smaller, the crossover point is taken to be the mean
          of the distribution: for this we use the approximation due to: R. Chattamvelli
          and R. Shanmugam, "Algorithm AS 310: Computing the Non-Central Beta
          Distribution Function", Applied Statistics, Vol. 46, No. 1. (1997),
          pp. 146-156.
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref3.svg"></span>

          </p></blockquote></div>
<p>
          Then either the CDF or its complement is computed using the relations:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref4.svg"></span>

          </p></blockquote></div>
<p>
          The summation is performed by starting at i = λ/2, and then recursing in
          both directions, using the usual recurrence relations for the Poisson PDF
          and incomplete beta functions. This is the "Method 2" described
          by:
        </p>
<p>
          Denise Benton and K. Krishnamoorthy, "Computing discrete mixtures
          of continuous distributions: noncentral chisquare, noncentral t and the
          distribution of the square of the sample multiple correlation coefficient",
          Computational Statistics &amp; Data Analysis 43 (2003) 249-267.
        </p>
<p>
          Specific applications of the above formulae to the noncentral beta distribution
          can be found in:
        </p>
<p>
          Russell V. Lenth, "Algorithm AS 226: Computing Noncentral Beta Probabilities",
          Applied Statistics, Vol. 36, No. 2. (1987), pp. 241-244.
        </p>
<p>
          H. Frick, "Algorithm AS R84: A Remark on Algorithm AS 226: Computing
          Non-Central Beta Probabilities", Applied Statistics, Vol. 39, No.
          2. (1990), pp. 311-312.
        </p>
<p>
          Ming Long Lam, "Remark AS R95: A Remark on Algorithm AS 226: Computing
          Non-Central Beta Probabilities", Applied Statistics, Vol. 44, No.
          4. (1995), pp. 551-552.
        </p>
<p>
          Harry O. Posten, "An Effective Algorithm for the Noncentral Beta Distribution
          Function", The American Statistician, Vol. 47, No. 2. (May, 1993),
          pp. 129-131.
        </p>
<p>
          R. Chattamvelli, "A Note on the Noncentral Beta Distribution Function",
          The American Statistician, Vol. 49, No. 2. (May, 1995), pp. 231-234.
        </p>
<p>
          Of these, the Posten reference provides the most complete overview, and
          includes the modification starting iteration at λ/2.
        </p>
<p>
          The main difference between this implementation and the above references
          is the direct computation of the complement when most efficient to do so,
          and the accumulation of the sum to -1 rather than subtracting the result
          from 1 at the end: this can substantially reduce the number of iterations
          required when the result is near 1.
        </p>
<p>
          The PDF is computed using the methodology of Benton and Krishnamoorthy
          and the relation:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span>

          </p></blockquote></div>
<p>
          Quantiles are computed using a specially modified version of <a class="link" href="../../roots_noderiv/bracket_solve.html" title="Bracket and Solve Root">bracket
          and solve</a>, starting the search for the root at the mean of the distribution.
          (A Cornish-Fisher type expansion was also tried, but while this gets quite
          close to the root in many cases, when it is wrong it tends to introduce
          quite pathological behaviour: more investigation in this area is probably
          warranted).
        </p>
</div>
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      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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